CATENARY AND ITS PRACTICAL APPLICATION IN CONSTRUCTION

The application of function graphs in architecture is closely related to the use of mathematical curves and surfaces to create both aesthetically pleasing and structurally efficient forms. They are key tools in engineering calculation and modern parametric design. An important place is occupied by algorithmic design, where the shape of an object is determined using a set of rules (algorithm) and mathematical dependencies between parameters. Instead of directly modeling geometry, the designer creates a logical structure that generates it, using mathematical functions (linear, polynomial, trigonometric, logarithmic, etc.) to calculate the values of one parameter based on others. Catenoids and catenary lines (or catenary curves) are important mathematical concepts that find applications, in particular, in architecture and engineering due to their unique properties related to the minimization of energy and stresses. The work is devoted to the analysis of the geometric and mathematical features of the catenoid (catenary line). The theoretical and practical aspects and the natural and practical application of the catenary line in various technical areas of activity are considered. A mathematical description of the catenoid is presented and its geometric and structural properties are investigated, which allow the use of this form in various technical areas, in particular in engineering, bridge construction, and in ancient and modern street architecture. The study noted that the catenary is a key curve in suspension bridge engineering (the overhanging form), and the inverted catenary (the chain arch) is in the architecture of arches and vaults (both in bridges and in elements that may be part of large fountain complexes), ensuring optimal material efficiency and structural strength. A conclusion was made about the possibility of expanding the scope of catenary applications